Spontaneous self-assembly of nanostructures has been a problem of long-standing interest for fabricating advanced electrical and optical devices. One approach towards controlled self-assembly is epitaxial growth on topographically patterned substrates, where surface features act as preferred sites for island formation. The substrate shape addressed here is a periodic array of rectangular mesas. Strained-layer growth on such substrates has been observed to form as islands on the edge, in the center, or at a combination of those locations on the top of the mesa. The purpose of this investigation is to discern whether these morphologies are driven by energetic or kinetic mechanisms. An energetic analysis is done under the assumption that the system free energy consists of surface free energy and strain energy. Strain arises due to a lattice mismatch between the film and substrate materials. The film and substrate are treated as two-dimensional linearly elastic solids with similar elastic properties. Under the constraint of fixed volume, various island arrangements and shapes are compared to determine low energy configurations. Islands are assumed to have the shape of a circular arc, and island positions at the center of the mesa and on the mesa edges are considered for one, two, and three island systems. For symmetric configurations, depending on mismatch strain, surface energy, volume, and relative substrate dimensions, it is found that either the configuration with a single centered island or the configuration with two edge mounted islands may be favored. Allowing for asymmetry, a single island placed on one edge of the substrate mesa is generally the low energy configuration. This occurs because islands located near the edge of a raised mesa are most efficient at relaxing mismatch strain due to an increased compliance. Kinetic simulations of morphological evolution during combined deposition and surface diffusion are also conducted in order to identify possible metastable or slowly evolving non-equilibrium states.